This document is a 3-page problem set for a class due on July 7th. It contains 4 problems involving concepts like preference relations, rationality of voters' preferences, and determining the likelihood of observing rational outcomes from random decisions. Solutions will be posted after class and there will be a test on the content on July 14th. Group work is allowed without limits on group size.

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Problem Set 1: Due in class on Tuesday July 7.
Solution
s to this homework will be posted
right after class, hence no late submissions will be accepted.
Test 1 on the content of this
homework will be given on Tuesday July 14 at 9:00am sharp.
Group solutions are welcomed
and encouraged. (There is no limit on the group size.)
Problem 1 (20p)
Figure 1 Figure 2

2. Figure 3 Figure 4
Figure 5 Figure 6

3. Figure 7 Figure 8
Figure 9 Figure 10
Figures 1-10 above depict preference and indifference relations
of ten different people on a
set of three or four alternatives. (For each decision maker her
set of alternatives constitutes

4. the entire domain of choice.) Alternatives are marked as dots
and preferences are marked
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with arrows: the direction of an arrow is the direction of the
preference relation, no arrow
between two dots means that an individual is indifferent
between these two alternatives.
In each of the ten figures preference relation either is or is not
transitive and also,
independently of the transitivity of the preference relation, the
indifference relation is or is
not transitive.
In the following table put “true” or “false” in each of the 20

5. empty cells.
Preference relation is
transitive
Indifference relation is
transitive
Figure 1
Figure 2
Figure 3
Figure 4
Figure 5
Figure 6
Figure 7

6. Figure 8
Figure 9
Figure 10
Problem 2 (10p)
Assume a voter has a strict preference over any two candidates
in the set of Clinton, Obama
and Edwards; in other words, it is not the case that he is
indifferent between some two
candidates in this set. In addition, assume that the voter is not
rational.
(1) Prove that a voter who prefers Obama over Clinton has to
prefer Edwards over Obama
and Clinton over Edwards. Present your reasoning.

7. (2) Prove that if all voters are not rational then either (i) all of
them have identical
preferences or (ii) the set of all voters is split into two subsets
such that in each subset all
voters have identical preferences.
Problem 3 (8p)
Let’s see now how likely would we observe a rational outcome
if a decision maker acted
randomly. Consider the case of three alternatives X, Y and Z.
Assume that deciding
between any two of these alternatives, e.g., X and Y, the
decision maker rolls a die and
dependin
each with equal probability.
(More specifically, assume that a decision maker takes any two
alternatives, say X and Y,

8. and rolls a die; if one or two dots come up he decides that he is
indifferent between X and Y,
if three or four dots come up he decides that he prefers X over
Y and if five or six dots come
up he decides that he prefers Y over X. Then he does the same
for X and Z and for Y and
Z.)
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ADVICE/HIT: Write out all possible outcomes. Each outcome
should be depicted as a
graph with three dots corresponding to X, Y and Z and arrows,
or lack thereof, between any
two dots. Now, each graph depicts a rational or a non-rational
preference system. Which

9. graphs correspond to rational systems? What is the relative
frequency of the rational ones in
the set of all systems? Note that this frequency is the
probability that individual’s
preferences determined at random, as described above, will end
up being rational?
Problem 4 (3 extra credit points)
Let’s go back to explaining the survey data in which 60% chose
Edwards over Obama, 60%
chose Edwards over Clinton, 60% chose Clinton over Obama
and then when asked to cast a
single vote for one of the three candidates, 45% chose Clinton,
30% Obama, and 25%
Edwards.

10. (1) Prove that it is not possible that all voters are rational (have
transitive preferences.)
(2) Suppose now that the distribution of votes was not 45%
chose Clinton, 30% Obama, and
25% Edwards but 38% Clinton, 32% Obama and 30% Edwards;
pair-wise data remains
unchanged. Prove that it is possible now that all voters are
rational (have transitive
preferences) and calculate the percentage of votes for each of
the six different orderings of
the three candidates.
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Problem Set 1: Due in class on Tuesday July 7.